Question: $f(x, y) = (\sin(xy), x^2)$ What is the curl of $f$ at $(3, \pi)$ ?
Explanation: The formula for curl in two dimensions is $\text{curl}(f) = \dfrac{\partial Q}{\partial x} - \dfrac{\partial P}{\partial y}$, where $P$ is the $x$ -component of $f$ and $Q$ is the $y$ -component. Let's differentiate! $\begin{aligned} \dfrac{\partial Q}{\partial x} &= \dfrac{\partial}{\partial x} \left[ x^2 \right] \\ \\ &= 2x \\ \\ \dfrac{\partial P}{\partial y} &= \dfrac{\partial}{\partial y} \left[ \sin(xy) \right] \\ \\ &= x\cos(xy) \end{aligned}$ Therefore: $\text{curl}(f) = 2x - x\cos(xy)$ The curl of $f$ at $(3, \pi)$ is $9$.